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Carrying Out a Test for the Slope of a Regression Model
This note page details the execution phase of a hypothesis test for the slope of a population regression line. This test helps us determine if there is a statistically significant linear relationship between two quantitative variables in the population, based on a sample. For the initial steps of defining hypotheses, refer to Setting Up a Test for the Slope of a Regression Model.
Conditions for Inference for Regression Slope
Before performing any calculations, we must verify that the conditions for inference about the slope of a population regression line are met. These conditions are often summarized by the acronym LINER:
| Condition | Description | How to Check |
|---|---|---|
| Linear | The true relationship between the explanatory variable ( $ x $ ) and the response variable ( $ y $ ) is linear. | Examine the scatterplot of the data; it should appear roughly linear. Check the residual plot for no obvious pattern. |
| Independent | Individual observations are independent of each other. When sampling without replacement, the population size should be at least 10 times the sample size ( $ N \ge 10n $ ). | Assumed if data are from a random sample or randomized experiment. Check the $ 10% $ condition for sampling without replacement. |
| ****Normal | For any fixed value of $ x $ , the response variable $ y $ varies according to a Normal distribution. In practice, this means the residuals are approximately Normally distributed. | Construct a histogram or Normal probability plot of the residuals. Look for approximate symmetry and no strong skewness. |
| Equal | The standard deviation of the response variable $ \sigma $ is the same for all values of the explanatory variable $ x $ . In practice, this means the variability of the residuals should be roughly constant across the range of predicted values. | Examine the residual plot; the scatter of points around $ y=0 $ should be roughly the same for all $ x $ values (no fan shape). |
| Random | The data comes from a well-designed random sample or randomized experiment. | State how the data were collected (random sample, randomized experiment, etc.). |
Failure to meet these conditions can invalidate the results of the hypothesis test.
Test Statistic
If the conditions are met, we can calculate the test statistic for the slope, which follows a $ t $ -distribution with $ df = n-2 $ degrees of freedom.
The test statistic is given by: $$ t = \frac{b - \beta_0}{SE_b} $$ Where:
- $ b $ is the slope of the sample regression line (least-squares regression line). This is our point estimate for the population slope.
- $ \beta_0 $ is the hypothesized value for the population slope, as stated in the null hypothesis ( $ H_0: \beta = \beta_0 $ ). In most cases, we test $ H_0: \beta = 0 $ (no linear relationship), so $ \beta_0 = 0 $ .
- $ SE_b $ is the standard error of the slope, which measures the variability of the sample slope $ b $ from sample to sample. This value is typically provided in computer output for regression analysis.
The formula for $ SE_b $ is: $$ SE_b = \frac{s}{\sqrt{\sum (x_i - \bar{x})^2}} $$ Where $ s $ is the standard deviation of the residuals (also known as the root mean square error, $ s_e $ ), and the denominator measures the spread of the $ x $ values. These values are often found in Linear Regression Models output.
Calculating the p-value
Once the test statistic $ t $ is calculated, we use it to find the p-value. The p-value is the probability of observing a sample slope as extreme as, or more extreme than, the one obtained, assuming the null hypothesis is true.
The p-value is calculated using a $ t $ -distribution with $ df = n-2 $ degrees of freedom, where $ n $ is the number of data points.
- For a two-sided test ( $ H_a: \beta \ne \beta_0 $ ): $ p\text{-value} = 2 \cdot P(T > |t|) $
- For a one-sided test ( $ H_a: \beta > \beta_0 $ ): $ p\text{-value} = P(T > t) $
- For a one-sided test ( $ H_a: \beta < \beta_0 $ ): $ p\text{-value} = P(T < t) $
The p-value can be found using a $ t $ -distribution table or statistical software.
Conclusion in Context
The final step is to make a decision about the null hypothesis and interpret the results in the context of the problem.
-
Decision: Compare the p-value to the chosen significance level $ \alpha $ .
- If $ p\text{-value} < \alpha $ , we reject the null hypothesis ( $ H_0 $ ).
- If $ p\text{-value} \ge \alpha $ , we fail to reject the null hypothesis ( $ H_0 $ ).
-
Interpretation: State the conclusion in clear, non-technical language, referring back to the original question.
- If you reject $ H_0 $ : There is convincing statistical evidence (at the $ \alpha $ level) to suggest that there is a linear relationship between $ x $ and $ y $ (or that the population slope is significantly different from $ \beta_0 $ ). Specifically, address the direction of the relationship if $ H_a $ was one-sided or the sign of $ b $ is clear.
- If you fail to reject $ H_0 $ : There is not convincing statistical evidence (at the $ \alpha $ level) to suggest a linear relationship between $ x $ and $ y $ (or that the population slope is significantly different from $ \beta_0 $ ). This does not mean there is no relationship, just that we don’t have enough evidence from our sample to conclude one exists at the chosen significance level.
Remember that a statistically significant relationship does not necessarily imply a strong relationship, nor does it imply causation. Always consider Potential Problems with Sampling and Inference and Experiments when interpreting results.
For further exploration, consider Confidence Intervals for the Slope of a Regression Model to estimate the true population slope.